\(\def\N{\mathbb{N}}\) \(\def\Z{\mathbb{Z}}\) \(\def\Q{\mathbb{Q}}\) \(\def\R{\mathbb{R}}\) \(\def\C{\mathbb{C}}\) \(\def\H{\mathbb{H}}\) \(\def\6{\partial}\) \(\DeclareMathOperator\Res{Res}\) \(\DeclareMathOperator\M{M}\) \(\DeclareMathOperator\ord{ord}\) \(\DeclareMathOperator\const{const}\) \(\DeclareMathOperator{\arccosh}{arccosh}\) \(\DeclareMathOperator{\arcsinh}{arcsinh}\) \(\DeclareMathOperator\id{id}\) \(\DeclareMathOperator\rk{rk}\) \(\DeclareMathOperator\tr{tr}\) \(\def\pt{\mathrm{pt}}\) \(\DeclareMathOperator\colim{colim}\) \(\DeclareMathOperator\Hom{Hom}\) \(\DeclareMathOperator\End{End}\) \(\DeclareMathOperator\Aut{Aut}\) \(\let\Im\relax\DeclareMathOperator\Im{Im}\) \(\let\Re\relax\DeclareMathOperator\Re{Re}\) \(\DeclareMathOperator\Ker{Ker}\) \(\DeclareMathOperator\Coker{Coker}\) \(\DeclareMathOperator\Map{Map}\) \(\def\GL{\mathrm{GL}}\) \(\def\SL{\mathrm{SL}}\) \(\def\O{\mathrm{O}}\) \(\def\SO{\mathrm{SO}}\) \(\def\Spin{\mathrm{Spin}}\) \(\def\U{\mathrm{U}}\) \(\def\SU{\mathrm{SU}}\) \(\def\g{{\mathfrak g}}\) \(\def\h{{\mathfrak h}}\) \(\def\gl{{\mathfrak{gl}}}\) \(\def\sl{{\mathfrak{sl}}}\) \(\def\sp{{\mathfrak{sp}}}\) \(\def\so{{\mathfrak{so}}}\) \(\def\spin{{\mathfrak{spin}}}\) \(\def\u{{\mathfrak u}}\) \(\def\su{{\mathfrak{su}}}\) \(\def\cA{\mathcal{A}}\) \(\def\cB{\mathcal{B}}\) \(\def\cC{\mathcal{C}}\) \(\def\cD{\mathcal{D}}\) \(\def\cE{\mathcal{E}}\) \(\def\cF{\mathcal{F}}\) \(\def\cG{\mathcal{G}}\) \(\def\cH{\mathcal{H}}\) \(\def\cI{\mathcal{I}}\) \(\def\cJ{\mathcal{J}}\) \(\def\cK{\mathcal{K}}\) \(\def\cL{\mathcal{L}}\) \(\def\cM{\mathcal{M}}\) \(\def\cN{\mathcal{N}}\) \(\def\cO{\mathcal{O}}\) \(\def\cP{\mathcal{P}}\) \(\def\cQ{\mathcal{Q}}\) \(\def\cR{\mathcal{R}}\) \(\def\cS{\mathcal{S}}\) \(\def\cT{\mathcal{T}}\) \(\def\cU{\mathcal{U}}\) \(\def\cV{\mathcal{V}}\) \(\def\cW{\mathcal{W}}\) \(\def\cX{\mathcal{X}}\) \(\def\cY{\mathcal{Y}}\) \(\def\cZ{\mathcal{Z}}\) \(\def\al{\alpha}\) \(\def\be{\beta}\) \(\def\ga{\gamma}\) \(\def\de{\delta}\) \(\def\ep{\epsilon}\) \(\def\ze{\zeta}\) \(\def\th{\theta}\) \(\def\io{\iota}\) \(\def\ka{\kappa}\) \(\def\la{\lambda}\) \(\def\si{\sigma}\) \(\def\up{\upsilon}\) \(\def\vp{\varphi}\) \(\def\om{\omega}\) \(\def\De{\Delta}\) \(\def\Ka{{\rm K}}\) \(\def\La{\Lambda}\) \(\def\Om{\Omega}\) \(\def\Ga{\Gamma}\) \(\def\Si{\Sigma}\) \(\def\Th{\Theta}\) \(\def\Up{\Upsilon}\) \(\def\Chi{{\rm X}}\) \(\def\Tau{{T}}\) \(\def\Nu{{\rm N}}\) \(\def\op{\oplus}\) \(\def\ot{\otimes}\) \(\def\t{\times}\) \(\def\bt{\boxtimes}\) \(\def\bu{\bullet}\) \(\def\iy{\infty}\) \(\def\longra{\longrightarrow}\) \(\def\an#1{\langle #1 \rangle}\) \(\def\ban#1{\bigl\langle #1 \bigr\rangle}\) \(\def\llbracket{{\normalsize\unicode{x27E6}}} \def\rrbracket{{\normalsize\unicode{x27E7}}} \) \(\def\lb{\llbracket}\) \(\def\rb{\rrbracket}\) \(\def\ul{\underline}\) \(\def\ol{\overline}\)

11  Applications to integrals

The residue theorem can be used to compute integrals of functions that do not have an obvious primitive. For this, we first need to find a suitable holomorphic function and contour to which we can apply the residue theorem, which is guesswork. The original integral is often a part of this contour and one shows that the rest of contour integral tends to zero.

We give three examples that illustrate the general technique.

Example 11.1  

Covered in lectures. Check back once the chapter is concluded.














Integrals made only of trigonometric functions can often be evaluated using the residue theorem and the following method.

Example 11.2  

Covered in lectures. Check back once the chapter is concluded.














The following integral was already computed in calculus using polar coordinates in the plane.

Example 11.3  

Covered in lectures. Check back once the chapter is concluded.














References

Dowson, H. R. 1979. “Serge Lang, Complex Analysis (Addison-Wesley, 1977), Xi + 321 Pp., £1200.” Proceedings of the Edinburgh Mathematical Society 22 (1): 65–65. https://doi.org/10.1017/S0013091500027838.
Freitag, E., and R. Busam. 2009. Complex Analysis. Universitext. Springer Berlin Heidelberg. https://books.google.co.uk/books?id=3xBpS-ZKlgsC.
Howie, J. M. 2012. Complex Analysis. Springer Undergraduate Mathematics Series. Springer London. https://books.google.co.uk/books?id=0FZDBAAAQBAJ.
Jameson, G. J. O. 1987. “H. A. Priestley, Introduction to Complex Analysis (Oxford University Press, 1985), 197 Pp., £8.50.” Proceedings of the Edinburgh Mathematical Society 30 (2): 325–26. https://doi.org/10.1017/S0013091500028406.
Remmert, R. 1991. Theory of Complex Functions. Graduate Texts in Mathematics. Springer New York. https://books.google.co.uk/books?id=uP8SF4jf7GEC.
Thomson, B. S., J. B. Bruckner, and A. M. Bruckner. 2001. Elementary Real Analysis. Prentice-Hall. https://books.google.co.uk/books?id=6l_E9OTFaK0C.